So once I have the work permit, will they tell me how to apply for the NI number? Do you know how long that will take in terms of how long it could take to get an appointment and then getting the number? Will they need to hold onto my passport?
You don't get a work permit, because you do not have a work visa. What you will get is a settlement (spousal) visa that allows work without you actually needing a work permit (work permits are only issued to people applying for employer-sponsored work visas, in which case they will likely be issued an NI number with their visa).
'They' will not tell you how to apply for an NI number, because not everyone who applies for FLR(M) will need one or want one. It's up to you to decide whether to apply for one or not, and find out how to do so.
See here for how to apply:
https://www.gov.uk/apply-national-insurance-numberThey will not take your passport. The appointment is simply to check you have the right to work and are looking for work and to fill out the form for you.
I ask this because I should receive my work permit (if all goes well) by beginning of September but then we have a USA trip at the end of that month.... so I thought and am hoping I have enough time to get the NIN before leaving.
As mentioned, it is not a work permit, it's a settlement (spousal) visa. It can take a few weeks to get the NI number, but you can travel while you are waiting for it if you like.